HIV-1 infection cannot be cured as it persists in latently infected cells that are targeted neither by the immune system nor by available therapeutic approaches. Consequently, a lifelong therapy suppressing only the actively replicating virus is necessary. The latent reservoir has been defined and characterized in various experimental models and in human patients, allowing research and development of approaches targeting individual steps critical for HIV-1 latency establishment, maintenance, and reactivation. However, additional mechanisms and processes driving the remaining low-level HIV-1 replication in the presence of the suppressive therapy still remain to be identified and targeted. Current approaches toward HIV-1 cure involve namely attempts to reactivate and purge HIV latently infected cells (so-called "shock and kill" strategy), as well as approaches involving gene therapy and/or gene editing and stem cell transplantation aiming at generation of cells resistant to HIV-1. This review summarizes current views and concepts underlying different approaches aiming at functional or sterilizing cure of HIV-1 infection.